WitCH 120: Unanalytic Continuation

Ten years ago, Numberphile came out with a now famous/infamous video featuring physicist Tony Padilla, in which Padilla “proves” that 1 + 2 + 3 + … = -1/12. It was quickly followed by a second, “what we really mean” video and some rebuttal notes by Padilla. At the time, Burkard and I (too) softly hammered Numberphile for this, in one of our final Age columns. A few years later, Burkard hammered Numberphile much harder, in an early Mathologer video.* Now, Numberphile has come out with two new videos: The return of -1/12, featuring Berkeley mathematician Tony Feng; and Does -1/12 protect us from infinity, with Padilla.

I have my views, but this a WitCH (thus summarising my views). So it’s your job. The two new videos are below. Go for it.

*) See also my post defending Burkard’s uncollegial hammering of Numberphile, an earlier WitCH on Numberphile (and Too Much Woo), a post on my general dislike of mathematics documentaries and on video as a medium for maths ed, and a video containing my proof that 1 + 2 + 3 (no dots) equals -1/12 (48:45 and 55:00). 


50 Replies to “WitCH 120: Unanalytic Continuation”

  1. Those “rebuttal” notes of Padilla’s need a trigger warning.

    He makes leaps of logic that are quite obscene and then derides those who take him to task as “so called experts” (his quotes).

    This one is easy: What is the Crap? The question should be Who?

    (Or Whom? I never know which way to go on that one. I’m going to go with Who this time though).

  2. OK gripe #1 – just because a sequence (or series) has an infinite number of terms, that does not mean it converges to something.

    The series \frac{9}{10}+\frac{9}{100}+\frac{9}{1000}+... does converge and so it is reasonable to ask “to what?”

    The series 1+10+100+... does not converge, so the comparison to the first series is not a fair one and asking “to what does it converge” is a meaningless question (which is kind of made obvious by the answer \frac{-1}{9}.

    To say “infinity” is not a concept encountered in the “real world” and 0.99999... does not get to the “finish line” of being equal to 1 is not a valid (nor proper) argument. If you want to say something like that, the onus is on you to say WHY rather than on anyone else to explain why not.

    I need a cold drink and a lie down before watching video 2 I suspect…

  3. Be careful RedFive, those “obscene” leaps of logic have a familiar ring. Let me take a tour through the history of physics.
    Newton used \Delta x/\Delta t for velocity, but for instantaneous velocity this was 0/0. He just cancelled the zeroes and obtained the “correct” answer. This was famously ridiculed by Bishop Berkeley. Ultimately Cauchy came to the rescue by slowing down this transition, looking at what the average velocity was approaching as the time interval became shorter.
    In work on boundary value problems, the workhorse is an improper integral that is divergent, \int_C \frac{f(t)}{x-t}dt. To get an answer, delete the infinite bit around t=x, say delete from x-\epsilon_1 to x+\epsilon_2. Then we let \epsilon_1 and \epsilon_2 approach zero in a way that their ratio approaches 1. The negative infinity will cancel with the positive one, giving the “correct”, or at least usable, answer.
    The Dirac delta is a “function” used throughout physics that is zero everywhere except at 0 where it is infinite, infinite enough to make the area under its graph equal 1! The simplest approach to make it respectable is to treat it as a limit of sequence of appropriate smooth functions that approach infinity at 0 and approach 0 elsewhere, and catching the properties that develop on the way.
    The most audacious example is renormalization in Quantum Field Theory. Feynman’s approach gives infinite values. Here his solution is just to delete the infinite bit. It works; the results remaining are claimed to agree with experiment to many decimal digits. It’s a great mystery.
    In mathematics there is a long established theory of summability that converts divergent series to convergent ones, and it’s useful for Fourier series. Basically you slow down the adding of the terms by multiplying them by factors that make the series convergent. Then you let these fiddle factors all approach 1. Until now I thought that this approach cannot work for the aggressively divergent 1+2+3+\cdots. But Padilla seems to have cracked it, giving of all things, -1/12. This is exciting. The standard connection is to think of this series as a case of the series \zeta(z) = 1/1^z+1/2^z+1/3^z+\cdots, the case z=-1. Although this series is divergent at z=-1, it is convergent for Re(z)>1, and this domain can be extended by analytic continuation, around the singularity to z=-1, giving \zeta(-1) = -1/12. Padilla’s approach suggests that there might be a shortcut that avoids this route. And it smells like renormalization.

      1. It depends what you mean by an infinite series. For example, is 1+1/2+1/4+\cdots =2? The answer is yes if we define the sum of the series in the appropriate way. Is 1+2+3 +\cdots = -1/12? The answer is yes if we define the sum of the series using the Padilla summability method. It will be important to add this proviso when using this method, otherwise people will just think you are a nutter. That’s where Padilla is a little cavalier.

          1. Buddha 🙂

            Yes of course it does. I recall at teachers college the lecturer was convincing us that multiple choice questions were bad. He challenged us to come up with a question where the choice was unarguable. Some brave soul suggested 1+1=2 and this was immediately shut down by two words “Boolean algebra”.

            1. The world is now full of maniacs who regard the statement 1 + 1 =2 as not sufficiently open-minded.

              Tom, this is all bullshit. And I know you know it is all bullshit.

              1. Looks like we need to agree to disagree. One thing we do agree is that when teaching wee-uns we don’t confuse them with later developments.

                  1. A “+” sign is written on the blackboard. Someone asks what it means.
                    Primary student’s answer: It’s the symbol for the addition of natural numbers.
                    Fields medalist’s answer: It’s the symbol for the addition of natural numbers.
                    Semi-educated lecturer’s answer: What a stupid questions! Did you even learn anything about mathematics? Maybe you shouldn’t attend this lecture at all…

                    1. Is this a true story? If so, in what way was the lecturer “semi” educated?

                      If not, why do you believe it is likely to be something that happens?

                      A plus sign is a plus sign. Natural numbers, rational numbers, complex numbers, matrices, it doesn’t matter.

                    2. I didn’t take it as a true story. Why believe it happens? Because I linked to the goddam videos where it happened. Or, if you want to really piss people off, define “woman”.

                      The perversion of language is a bad disease, and there is zero excuse for mathematicians doing it.

    1. Yeah, this is what I’ve always thought about this dispute between, on one side, Burkard and Marty, and on the other side, Tony and Brady: it’s a dispute between mathematicians and physicists (well, one physicist) about the correct epistemology for mathematics. Physicists are well known for infuriating mathematicians with the way they play fast and loose around questions of “convergence” or “defining the space in which one is working”. But they often get results that can then be formalized with the help of some mathematical theory.

      The equation 1 + 2 + 3 + … = -1/12 is of course incorrect with the standard metric on the real numbers. But the point is that there are contexts in which we can assign this value to said sum, and in fact, it seems that -1/12 is the only “sensible” finite value that can be given to that sum. (Tony Padilla prefaces everything with his opinion that he doesn’t like infinity and he thinks it has no place in physics; that’s not really a scientific belief, but it colours the work that he does.) Does this mean that we can write 1 + 2 + 3 + … = -1/12? Given the correct metric, I guess that we could. Tony Padilla doesn’t provide this metric, which I guess is why this infuriates Marty.

      I don’t have much of an issue with the video featuring Tony Feng. It’s a typical Numberphile video where a graduate student or young professor presents either a theme related to their research (in the “my thesis in 360 seconds” mould) or some other mathematical subject that is of interest to a lay audience. This sort of video sometimes requires a few shortcuts, which I understand Marty dislikes, but you’d see this in any expository talk on any subject that’s intended for laypeople. I think that Tony Feng is careful to say that the series 1 + 2 + 3 + … does not converge in the classical sense, but that there is a way involving the Riemann zeta function to assign to it the value -1/12.

      I cannot really comment on the most recent video with Tony Padilla. He’s describing some of the impetus behind his current research, which doesn’t seem to be easily explainable to lay audiences. (I prefer the Numberphile videos where the explanation can fit within the video to those where the audience has to accept things without proof.) If it is the case that any way the sum 1 + 2 + 3 + … is regularized by a (suitable) weighing factor, it ends up asymptotically growing like C*N^2 – 1/12 plus subconstant terms, that seems to me to be an interesting result. But there is no real explanation why this doesn’t work when using the “sharp cutoff” weighing function.

      1. Thanks, Marc. I’m not bothered whatsoever by, much less infuriated by, physicists playing fast and loose with mathematics. The Feng video is not a perversion like the Padilla video, but it is obviously very bad, and it takes pure desperation to imagine otherwise.

      2. But after that Padilla regularizes with another weighting factor that eliminates the C N^{1/2} term, which is why I take it seriously.

          1. Ah, has there been a miscommunication? My submission “But after that …” was meant as a reply to Marc, because he seemed to have missed the later part of the Padilla video which was giving a new summability method that makes the dropping of the infinite part a little more palatable. But perhaps Marty took it as a reply to his “I’m not bothered …” post.

            I thought that this discussion was asking for general reactions to the two videos, but it seems that we are required to stick to the validity of the statement 1+2+3+\cdots = -1/12  ...(1). Sorry, I missed the earlier history. Let me make my personal preference clear.

            Hardy was a stickler for rigour having spent a career trying to raise the standard of British mathematics to the continental level. But when Ramanujan sent the famous letter that included equation (1) he made no complaint. Hardy would have been familiar with the analytic continuation connection, but presumably Ramanujan was not. (We may never know what Ramanujan had in mind.) Why was Hardy tolerant? I went looking in his book Divergent \ Series and found the following.
            “… but it is broadly true to say that mathematicians before Cauchy asked not ‘How shall we define 1-1+1-…?’ but ‘What is 1-1+1-…?’, and that this habit of mind led them into unnecessary perplexities and controversies which were often really verbal.”

            This does not mean the modern attitude gives permission for any old definition. We need to demonstrate that it makes some sense and is a useful extension of the standard definition. Padilla makes a reasonable case for extending infinite sums to include (1). But I personally would prefer not to go so far. When we use divergent series in asymptotics (where they are essential for practical computation) it’s standard to use the symbol \sim rather than =, to emphasise that the summation is not meant in the classical sense. Similarly I prefer to write
            1+2+3+\cdots \sim -1/12. This hopefully avoids confusing youngsters and vitriolic arguments with Marty.

            1. Tom, of course you are free to comment on any aspect of the Padilla video. But it’s pretty obtuse to so comment while considering the video in a vacuum: the Numberphile guys specifically noted the history and I specifically noted the history.

              If there had ever been an ounce of acknowledgment from either Padilla or Brady that they stuffed up, royally, then Padilla’s video could have been fine and good (for the genre). But they didn’t and it isn’t.

              You are, anyway, way too forgiving of their abuse of language and terminology. Such abuse always has consequences, and for videos watched by millions of people who don’t know better, the consequences are very large and very bad. Don’t excuse these smug clowns.

    2. OK Tom, this is going to be as concise as I can make it.

      I reject your premises and I reject your conclusion.

      If anyone wants to define the “metric” that makes 1+2+3+...=\frac{-1}{12} I’m listening, but suspect you will also need to redefine the word “sensible”.

      1. Well, the metric isn’t the problem. Just take the zero function, inducing the trivial topology on R (or C), which makes the series converge to -1/12. One might discuss the “=” sign then, of course, because the left hand side of the equation would no longer represent a unique number.

        1. I took “metric” to mean “scenario”, rather than a formal metric space thing. With “scenario”, I take Marc’s point. It simply doesn’t justify the abuse of language.

      2. Thanks Red Five. If you look closely you should find that I never wrote that I would use 1+2+3+\cdots = 1/12, rather that we can make definitions that would allow it. I have just replied to Marty clarifying my attitude. Perhaps I am still a little pre-Cauchy.

        PS It’s not a metric in any sense that I can see?

        1. I’ll accept that. I don’t think it changes much.

          “Metric” was used in a different comment. I got my wires crossed.

        1. Yes. I think it is Tao’s approach that Padilla generalised or polished or whatever. But, for the third time, and Padilla’s interesting work notwithstanding, none of this justifies Padilla’s/Numberphile’s bullshit.

          Language matters. Maybe only to me, it would appear. But I’m gonna keep waving that flag until I drop dead.

  4. There are different ways to sum an infinite series e.g. Cesaro summation. These different methods can lead to different answers. They are discussed in G.H. Hardy “Infinite series”.

  5. When I talked about a metric, I thought that there might be a (nontrivial) way to make the real numbers into a metric space such that 1 + 2 + \cdots would be in this space a converging series with the sum -\frac1{12}. It reminded me of p-adic numbers: if I look at the examples on the Wikipedia page (https://en.wikipedia.org/wiki/P-adic_number), I see that we get, using the 5-adic valuation, the expansion

        \[\tfrac13 = 2\times 5^0 + 3\times 5^1 + 1\times 5^2 + 3\times 5^3 + 1\times 5^4 + \cdots.\]

    Though I do notice that the above expression isn’t actually written on the Wikipedia page; several notations for p-adic numbers are discussed, but not this particular one, possibly because of concerns similar to Marty’s. This said, I wouldn’t have a problem with the above expression. The series on the right is a divergent series using the normal metric on the real numbers, but it is convergent with the 5-adic valuation since the 5^n term becomes arbitrarily small for this metric as n becomes large.

    I guess that there might not be a way to formally define a metric space in a way that 1 + 2 + \cdots = -\frac1{12} in this space. (Or is there, in a way that isn’t trivial and doesn’t make equality completely meaningless? That’s the thing with this, I still feel it should be feasible.) So that would make Tom right that 1 + 2 + \cdots is something -\frac1{12}, but that the something might be better written {}\sim{} rather than {}={}. Still, I don’t think I share Marty’s dislike of the use of the equality sign in this context. It has nothing to do with attempts to say that maybe 2 + 2 = 5, or to redefine what it is to be a woman (and these two latter cases, which Marty probably also interprets as attempts to muddy the meaning of the equality sign, are themselves completely different from each other).

    1. Saying 1 + 2 + 3 + … equals -1/12 perverts the meaning of language and discussion in a very similar way to saying a man who has had his dick chopped of is a woman. One can argue, in either case, whether the intent is well-meaning or not, but the effect in both cases is, without doubt, perverse.

  6. Saying 1+2+3… = -1/12 is the same as saying 0 = 1 + 12(1+2+3…). Both are false no matter which way you look at it. End of story. I’m not sure why this even needs debating. I’m concerned that it has gained so much exposure. It’s macadamia level nuts.

      1. I’m interested in the efforts to defend ideas such as this which I believe are clearly false. It can be a useful learning experience.

        Not expecting to be convinced it is correct but willing to read/listen.

        Also, what BiB said below.

      2. “The Bullshit Asymmetry Principle, also known as Brandolini’s Law, states that the amount of energy needed to refute bullshit is an order of magnitude bigger than that needed to produce it.”

        I disagree with Brandolini. It is at least two orders of magnitude more energy intensive, if not three. This Witch could be considered ‘Exhibit A’ or ‘Exhibit -1/12’. And the amount of energy expended should not be considered in any way as evidence that the b.s. has any merit whatsoever.

          1. One of the best quotes in history:

            “I know that most men—not only those considered clever, but even those who are very clever, and capable of understanding most difficult scientific, mathematical, or philosophic problems—can very seldom discern even the simplest and most obvious truth if it be such as to oblige them to admit the falsity of conclusions they have formed, perhaps with much difficulty—conclusions of which they are proud, which they have taught to others, and on which they have built their lives.”

            Leo Tolstoy, What Is Art? (1897)

              1. “The reluctance of people to call out this bullshit, and bullshit generally, is a wonder to behold.”

                — marty, WitCH 120 (2024)

                Tolstoy does come in handy, from time to time. And he speaks to all of us, if we would only listen.

                  1. My excuse is that it is better to skip a stone across a pond than it is to attempt to swim the English Channel. More specifically, the intransigent adherence to b.s. or, worse, the active promulgation of it as fact, almost always comes down to ego. That is, there is a strong positive correlation between ego and b.s. and Tolstoy describes this better than anyone I have read. Calling out b.s. can, therefore, be misconstrued as attacking the ego. I just hope that people consider this before they accuse others of attacking them, and also consider the possibility that they might be wrong. However, the burden of proof rests with the person making the claim and not with those who disagree with it, regardless of how ‘uneducated’ we may appear on first acquaintance.

                    1. I know tom, and it’s not ego. (Padilla, on the other hand …) But tom’s clearly interested in and invested in this stuff to the extent that it’s very difficult for him to see the point.

  7. \displaystyle - \frac{1}{12}?? What a crock. Everyone knows the answer is 42.

    (PS – Everyone also knows that it’s not about the mathematics, it’s about the ‘likes’ and the ‘clicks’ and the attempt to monetise everything).

  8. “Nope. Not having it. State your claim clearly and defend it, or admit defeat.”

    Dear Marty, I was assuming that I had won the argument because all I got from you was abuse:
    “The world is now full of maniacs who regard the statement 1 + 1 =2 as not sufficiently open-minded.”
    “Tom, this is all bullshit. And I know you know it is all bullshit.”
    Since you have not verbalised where the Padilla conclusion is wrong, I assume that your reaction is based on gut feel.

    I made the mistake earlier of giving examples of mathematics that originated in physics. Although these are all now part of mainstream mathematics, you dismissed them out of hand. So here I will stick to examples that arose in mathematics proper. This will sound pompous to a mathematician, but some of your audience clearly need educating.

    Starting with negative numbers. Medieval philosophers thought that numbers were only for counting or measuring so resisted the idea of negatives. The problem 2-5 had no answer. Even 2-2 left no length to measure. The sheer usefulness of negatives carried the day so people came to accept that 2-5 is \hbox{\em defined} as a new thing, -3. But note that in making new definitions we don’t have open slather. The new numbers must be both useful and not conflict with previous practice with counting numbers. Definition 1, gut feel 0.

    Infinite decimals. We now \hbox{\em define} an infinite decimal to be the limiting value of a truncated version as the length increases. So 333333\bar{3} is \hbox{\em defined} to be 1/3. If we must use decimal notation this is clearly useful. And the definition does agree with the arithmetic of fractions, in fact allows decimals of irrationals. You and Burkard had trouble convincing some people that 0.999999\bar{9} is 1. In mathematics it is \hbox{\em defined}, not proved, to be 1; presumably the dissenters have a gut feel that it must be something different.

    Hardy, in my earlier quotation, was happy to \hbox{\em define} infinite series to be their Cesaro, or Abel, summable results. So we have 1-1+1-1+\cdots = 1/2. This is useful in as much as it extends Fourier theory, so that a Fourier series gives a valid result more generally. And these types of summability always agree with the earlier limit-of-sum definition for infinite series, so there is no conflict. In my work I prefer to write “Cesaro summable to” rather than the equal sign, because there may be readers unfamiliar with the generally accepted new definition, and their guts may feel upset.

    So lets turn to the Padilla summability method. I am not happy with the equal sign in the statement 1+2+3+\cdots = -1/12. The derivation may turn out to be useful, giving a short cut to the complex plane and a whole new wonderful playground. But the statement does conflict with a long-standing principle that addition of positive numbers should give a positive result, if it gives anything. I guess all the unexplained rejection of the result by your correspondents is caused by this simple fact. Padilla will be going against a sea of gut feel and should have expected the response.

    You state: “I do not believe that you can give a precise and reasonable definition that ‘allows’ 1 + 2 + 3 + … = -1/12.”
    The precise bit is, according to the video, \sum_{n=1}^\infty a_n = \lim_{N\to\infty} \sum_{n=1}^\infty a_n \exp(-n/N)\cos(n/N) which is a simple extension of Abel summability (Terry Mills is not being irrelevant.) The question of reasonableness runs up against the fact that the sum of positive numbers “should” be positive. However, I am ready to drop this requirement if Padilla can show that the new calculus is very useful, so will check back in 2124.

    PS. This is called arguing using logic. You are often very good at it, but I guess you don’t have time to reply to every post in this way.

    1. *) Your first, second and final paragraphs:

      I could make you an offer: I’ll be less aggressive if you’ll be less sanctimonious. But I’m not offering.

      I haven’t verbalised where Padilla is wrong? Are you joking? He is wrong to declare that 1 + 2 + 3 + … = -1/12. Padilla is simply not allowed to make the dots mean whatever he wishes, ignoring all accepted meaning.

      *) Third paragraph:

      Thanks for the history lesson. Not really relevant, but fine.

      *) Fourth paragraph:

      No, 0.3333… is not defined to be 1/3. No, 0.9999… is not defined to be 1. If you’re gonna give us littlies, who so “need educating”, a lesson, you might want to get these things right.

      Yes, Burkard and I have made a crusade of convincing students and teachers and the public that 0.9999… = 1. It is a hell of harder to do that when the clowns like Padilla spout his nonsense to millions of people. Seriously, the damage that Padilla and Brady did with that video is monumental. Did the video also do some good? Yes. But not nearly as much good as bad.

      *) Fifth paragraph:

      Of course there is value in extending definitions of summability beyond standard convergence. Of course Cesaro and Abel and, I’m assuming, Padilla, “summability” is interesting and important. But the point, the only point, the point you make in this paragraph, is one must state absolutely clearly the notion of summability one is using. You do not, ever, use established notation and blithely steal it for your own uses, at least not without neon signs that that is what you’re doing. Well, maybe if you’re Hardy or Ramanujan or Euler you do: geniuses have license that mortals do not. But not if you’re a smug knowitall clown on A YouTube show. This is the only point, but it is a huge point.

      And what, really, is Padilla’s purpose in obfuscating in this manner, other than to be a clever dick? He can do exactly the work he is doing, and piss off absolutely no one, and please some people with some nice maths and his genuine intelligence and hard work, simply by ceasing to be a clever dick.

      *) Sixth paragraph:

      You gave a precise definition so that 1 + 2 + 3 + … = -1/12. It doesn’t look very reasonable to me, even looking no further. And then, even ignoring the contradiction with accepted meaning, how do you define other series? What, for instance, is the definition of 1^2 + 2^2 + 3^2 + …? The Indians didn’t make sense of -5: they made sense of -n.

  9. Given \displaystyle  1+2+3+\cdots = -1/12, what happens when we start subtracting numbers from both sides …?

    I hope 1 is still equal to 1, 1 + 2 is still equal to 3 etc.

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